Abstract
We study the dynamics of a one-predator, two-prey system in which the predator has an indirect effect on the preys. We show that, in presence of the indirect effect term, the system admits coexistence of the three populations while, if we disregard it, at least one of the populations goes to extinction.
Highlights
The importance of indirect effects is well established in Biology, for example, in the case of predation, the predator can alter the morphology or the behavior of the preys
We study the dynamics of a one-predator, two-prey system in which the predator has an indirect effect on the preys
We have shown that in absence of the terms that describe indirect effects, the system (5) does not admit coexistence of the three populations
Summary
The importance of indirect effects is well established in Biology (see [1,2,3,4,5]), for example, in the case of predation (see [6]), the predator can alter the morphology (see [7]) or the behavior of the preys. The population of predator Z has an effect on the prey C which is described by the term −mCZ which produce a negative growth rate This effect produces an indirect effect that is positive (the term mCZ). In this region we have that C = 0, that is, the plane C = 0 is invariant and as a consequence we have only to check the vector field on the boundary of T2. We sum the equations of the system, we obtain the following differential equation depending on V: V = (I0 − V) (acC + agG) − eZ This concludes the proof since if the solutions start on T4 they enter the set Δ or they remain on T4.
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